We follow the proof of Proposition 4.7 in [1]}.
We first prove for \(k=1\) , and writing \(\lambda ,E,m\)  instead of \(\lambda _{i_1},E_{i_1},m_{i_1}\) .
Let \(\psi (\omega ,x):=\log \det (Df_{\omega }(x)|E(\omega ,x))\)  for \((\omega ,x)\in \Gamma \) .
It follows from Lemma REF  immediately that \(\psi \)  is measurable.
The Proposition REF  (3) tells us
\(\psi (\omega ,x)\le m\log m^{\frac{1}{2}} |f_{\omega }|_{B(A_{\omega },r_0)}|_{C^2}.\) 
So, by assumption (H2) (iii) one has
\(\psi ^{+}(\omega ,x)\in L^1({u})\) .
Using the Birkhoff ergodic theorem, there is a constant \(\gamma \in [-\infty ,\infty )\)  such that for \(\mu \) -almost every \((\omega ,x)\) ,
\( \gamma :=\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{i=0}^{n-1} \psi (\Phi ^{i}(\omega ,x))=\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{i=0}^{n-1} \psi (\Phi ^{-i}(\omega ,x)).\) 
It suffices to show that \(\gamma =m\lambda \) .
On the one hand, by Proposition REF  (3) we choose a unit basis \(\lbrace v_i\rbrace _{i=1}^{m}\)  for \(E(\omega ,x)\)  such that
\(\det (Df^{n}_{\omega }(x)|E(\omega ,x))\le m^{m/2} \prod _{i=1}^{m}|Df^{n}_{\omega }(x)v_i|.\) 
By Theorem REF , we have that \(\gamma \le m\lambda \) .
On the other hand, note that
\(\sum _{i=0}^{n-1} \psi (\Phi ^{-i}(\omega ,x))&=\det (Df^{n}_{\theta ^{-n}\omega }(f_{\omega }^{-n}x)|E(\Phi ^{-n}(\omega ,x)))\\&=(\det (Df^{-n}_{\omega }(x)|E(\omega ,x)))^{-1}.\) 

We also choose a unit basis \(\lbrace u_i\rbrace _{i=1}^{m}\)  for \(E(\omega ,x)\)  such that
\(\det (Df^{-n}_{\omega }(x)|E(\omega ,x))\le m^{m/2} \prod _{i=1}^{m}|Df^{-n}_{\omega }(x)u_i|.\) 
Also by Theorem REF , we have that \(\gamma \ge m\lambda \) .
This proves the case when \(k=1\) .
Proceeding to the general case, it suffices to prove the case \(k=2\) . By Proposition REF , we have
\(\dfrac{1}{C_{m_{i_1}+m_{i_2}}\cdot |\pi _{i_1}(\Phi ^{n}(\omega ,x))|^{m_{i_1}}}&\le \dfrac{\det (Df^{n}_{\omega }(x)|E_{i_1}(\omega ,x)\oplus E_{i_2}(\omega ,x))}{\det (Df^{n}_{\omega }(x)|E_{i_1}(\omega ,x))\det (Df^{n}_{\omega }(x)|E_{i_2}(\omega ,x))}\\&\le C_{m_{i_1}+m_{i_2}}|\pi _{i_1}(\omega ,x)|^{m_{i_1}}.\) 

Recall that Theorem REF  tells us \(|\pi _{i_1}|\)  is \(\Phi \) -temperate.
So, together with the case \(k=1\) , the case for \(k=2\)  is also valid.
